The complementary polynomials and the Rodrigues operator. A distributional study

Abstract

We can write the polynomial solution of the second order linear differential equation of hypergeometric-type $$\varphi (x)y^{\prime \prime }+\psi (x)y^{\prime }+\lambda y=0,$$ where $\varphi$ and $\psi$ are polynomials, $°\varphi \le 2$, $°\psi =1$ and $\lambda$ is a constant, among others, by using the Rodrigues operator $R_k(\varphi ,\mathbf{u})$ (see \cite{coma2}) where $\mathbf{u}$ is certain linear operator which satisfies the distributional equation \begin{equation} \label{1} \frac{d}{dx}[ϕ{\bf u}]=ψ{\bf u}, \end{equation} as $$P_n(x)=B_n{R}_n(\varphi ,\mathbf{u})[1],B_n\ne 0,n=0,1,2,...$$ Taking this into account we construct the complementary polynomials. Among the key results is a generating functional function in closed form leading to derivations of recursion relations and addition theorem. The complementary polynomials satisfy a hypergeometric-type differential equation themselves, have a three-term recursion among others and Rodrigues formulas.